Abstract

This paper considers dynamical behaviors of a class of fuzzy
impulsive reaction-diffusion delayed cellular neural networks
(FIRDDCNNs) with time-varying periodic self-inhibitions,
interconnection weights, and inputs. By using delay differential
inequality, -matrix theory, and analytic methods, some new
sufficient conditions ensuring global exponential stability of the
periodic FIRDDCNN model with Neumann boundary conditions are
established, and the exponential convergence rate index is
estimated. The differentiability of the time-varying delays is not
needed. An example is presented to demonstrate the efficiency and
effectiveness of the obtained results.

1. Introduction

The fuzzy cellular neural networks (FCNNs) model, which combines fuzzy logic with the structure of traditional neural networks (CNNs) [1–3], has been proposed by Yang et al. [4, 5]. Unlike previous CNNs structures, the FCNNs model has fuzzy logic between its template and input and/or output besides the “sum of product” operation. Studies have shown that the FCNNs model is a very useful paradigm for image processing and pattern recognition [6–8]. These applications heavily depend on not only the dynamical analysis of equilibrium points but also on that of the periodic oscillatory solutions. In fact, the human brain is naturally in periodic oscillatory [9], and the dynamical analysis of periodic oscillatory solutions is very important in learning theory [10, 11], because learning usually requires repetition. Moreover, an equilibrium point can be viewed as a special periodic solution of neural networks with arbitrary period. Stability analysis problems for FCNNs with and without delays have recently been probed; see [12–22] and the references therein. Yuan et al. [13] have investigated stability of FCNNs by linear matrix inequality approach, and several criteria have been provided for checking the periodic solutions for FCNNs with time-varying delays. Huang [14] has probed exponential stability of fuzzy cellular neural networks with distributed delay, without considering reaction-diffusion effects.

Strictly speaking, reaction-diffusion effects cannot be neglected in both biological and man-made neural networks [19–32], especially when electrons are moving in noneven electromagnetic field. In [19], stability is considered for FCNNs with diffusion terms and time-varying delay. Wang and Lu [20] have probed global exponential stability of FCNNs with delays and reaction-diffusion terms. Song and Wang [21] have studied dynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coefficients and delays without considering pulsing effects. Wang et al. [22] have discussed exponential stability of impulsive stochastic fuzzy reaction-diffusion Cohen-Grossberg neural networks with mixed delays. Zhao and Mao [30] have investigated boundedness and stability of nonautonomous cellular neural networks with reaction-diffusion terms. Zhao and Wang [31] have considered existence of periodic oscillatory solution of reaction-diffusion neural networks with delays without fuzzy logic and impulsive effect.

As we all know, many practical systems in physics, biology, engineering, and information science undergo abrupt changes at certain moments of time because of impulsive inputs [33]. Impulsive differential equations and impulsive neural networks have been received much interest in recent years; see, for example, [34–42] and the references therein. Yang and Xu [36] have investigated existence and exponential stability of periodic solution for impulsive delay differential equations and applications. Li and Lu [38] have discussed global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses without reaction-diffusion. To the best of our knowledge, few authors have probed the existence and exponential stability of the periodic solutions for the FIRDDCNN model with variable coefficients, and time-varying delays. As a result of the simultaneous presence of fuzziness, pulsing effects, reaction-diffusion phenomena, periodicity, variable coefficients and delays, the dynamical behaviors of this kind of model become much more complex and have not been properly addressed, which still remain important and challenging.

Motivated by the above discussion, we will establish some sufficient conditions for the existence and exponential stability of periodic solutions of this kind of FIRDDCNN model, applying delay differential inequality, -matrix theory, and analytic methods. An example is employed to demonstrate the usefulness of the obtained results.

Notations. Throughout this paper, and denote, respectively, the -dimensional Euclidean space and the set of all real matrices. The superscript “” denotes matrix transposition and the notation (resp., ), where and are symmetric matrices, means that is positive semidefinite (resp., positive definite). is a bounded compact set in space with smooth boundary and measure ; Neumann boundary condition is the outer normal to ; is the space of real functions which are for the Lebesgue measure. It is a Banach space with the norm , where ,, , . For function with positive period , we denote . Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.

2. Preliminaries

Consider the impulsive fuzzy reaction-diffusion delayed cellular neural networks (FIRDDCNN) model:
where is the number of neurons in the network and corresponds to the state of the th neuron at time and in space ; is the diffusion-matrix and ; is the Laplace operator; denotes the activation function of the th unit and the activation function of the th unit; is an input at time ; represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the networks and external inputs at time ; and are elements of feedback template and feed forward template at time , respectively. Moreover, in model (1), , and are elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed forward MIN template, and fuzzy feed forward MAX template at time , respectively; the symbols “" and “" denote the fuzzy AND and fuzzy OR operation, respectively; time-varying delay is the transmission delay along the axon of the th unit and satisfies ( is a constant); the initial condition is bounded and continuous on , where . The fixed moments satisfy . and denote the right-hand and left-hand limits at , respectively. We always assume , for all . The initial value functions belong to . for every ; for any fixed is continuous for all but at most countable points and at these points, and exist, , where and denote the right-hand and left-hand limit of the function , respectively. Especially, let . For any , suppose that exists as a finite number and introduce the norm , where .

Throughout the paper, we make the following assumptions. There exists a positive diagonal matrix , and such that
for all ., and are periodic function with a common positive period for all . For , there exists such that , and are Lipschitz continuous in .

Definition 1. The model in (1) is said to be globally exponentially periodic if (i) there exists one -periodic solution and (ii) all other solutions of the model converge exponentially to it as .

Definition 2 (see [26]). Let , where and . Then the function is called an -function, if (i) for every , there holds , for , where and ; (ii) every th element of satisfies for any , where arbitrary and belong to and have the same th component . Here, .

Definition 3 (see [26]). A real matrix is said to be a nonsingular -matrix if ; and all successive principal minors of are positive.

Lemma 4 (see [13]). Let and be two states of the model in (1), then we have

3. Main Results and Proofs

We should first point out that, under assumptions (H1), (H2), and (H3), the FIRDDCNN model (1) has at least one -periodic solution of [26]. The proof of the existence of the -periodic solution of (1) can be carried out similar to [26, 28] by the nonlinear functional analysis methods such as topological degree and here is omitted. We will mainly discuss the uniqueness of the periodic solution and its exponential stability.

Theorem 6. Assume that – holds. Furthermore, assume that the following conditions hold is a nonsingular -matrix. The impulsive operators is Lipschitz continuous in ; that is, there exists a nonnegative diagnose matrix such that for all , , where , ,., where .

Then the model (1) is global exponential periodic and the exponential convergence rate index and can be estimated by
where and , , , , satisfies .

Proof. For any , let be a periodic solution of the system (1) starting from and , a solution of the system (1) starting from . Define
and we can see that for all . Let , then from (1) we get
for all , , .Multiplying both sides of (6) by and integrating it in , we have
for , , . By boundary condition and Green Formula, we can get
Then, from (8), (9), (H1)-(H2), Lemma 4, and the Holder inequality,
Thus,
for . Since is a nonsingular -matrix, there exists a vector such that
Considering functions
we know from (11) that and is continuous. Since , is strictly monotonically increasing, there exists a scalar such that
Choosing , we have
That is,
Furthermore, choose a positive scalar large enough such that
For any , let
From (15)–(17), we obtain
where and . It is easy to verify that is an -function. It follows also from (16) and (17) that
Denote
where , then
From (10), we can obtain
Now, it follows from (18)–(22) and Lemma 5 that
Letting , we have
And moreover, from (24), we get
Let , then . Define ; it follows from (25) and the definitions of and that
It is easily observed that
Because (26) holds, we can suppose that for inequality
holds, where . When , we note (H5) that
where is the spectral radius of . Let , by (28), (29), and , we obtain
Combining (10), (17), (30), and Lemma 5, we get
Applying mathematical induction, we conclude that
From (H6) and (32), we have
This means that
choosing a positive integer such that
Define a Poincare mapping by
Then
Setting in (34), from (35) and (37), we have
which implies that is a contraction mapping. Thus, there exists a unique fixed point such that
From (37), we know that is also a fixed point of , and then it follows from the uniqueness of the fixed point that
Let be a solution of the model (1), then is also a solution of the model (1). Obviously,
for all . Hence, , which shows that is exactly one -periodic solution of model (1). It is easy to see that all other solutions of model (1) converge to this periodic solution exponentially as , and the exponential convergence rate index is . The proof is completed.

Remark 7. When , , , , , , , , , and , and are constants), then the model (1) is changed into
For any positive constant , we have , , , , , , , , , and for . Thus, the sufficient conditions in Theorem 6 are satisfied.

Remark 8. If , the model (1) is changed into
which has been discussed in [22]. As Song and Wang have pointed out, the model (43) is more general than some well-studied fuzzy neural networks. For example, when , and are all constants, the model in (43) reduces the model which has been studied by Huang [19]. Moreover, if , , , , then model (42) covers the model studied by Yang et al. [4, 5] as a special case. If and is assumed to be differentiable for , then model (43) can be specialized to the model investigated in Liu and Tang [12] and Yuan et al. [13]. Obviously, our results are less conservative than that of the above-mentioned literature, because they do not consider impulsive effects.

4. Numerical Examples

Example 9. Consider a two-neuron FIRDDCNN model:
where , , , , , , , , , , , , , , , , , , , , , , , , , , , , . We assume that there exists such that . Obviously, , and satisfy the assumption (H1) with and (H2) and (H3) are satisfied with a common positive period
is a nonsingular -matrix. The conditions of Theorem 6 are satisfied, hence there exists exactly one -periodic solution of the model and all other solutions of the model converge exponentially to it as . Furthermore, the exponential converging index can be calculated as , because here and . The simulation results are shown in Figures 1, 2, 3, and 4, respectively.

5. Conclusions

In this paper, periodicity and global exponential stability of a class of FIRDDCNN model with variable both coefficients and delays have been investigated. By using Halanay’s delay differential inequality, -matrix theory, and analytic methods, some new sufficient conditions have been established to guarantee the existence, uniqueness, and global exponential stability of the periodic solution. Moreover, the exponential convergence rate index can be estimated. An example and its simulation have been given to show the effectiveness of the obtained results. In particular, the differentiability of the time-varying delays has been removed. The dynamic behaviors of fuzzy neural networks with the property of exponential periodicity are of great importance in many areas such as learning systems.

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and constructive suggestions. This research is supported by the Natural Science Foundation of Guangxi Autonomous Region (no. 2012GXNSFBA053003), the National Natural Science Foundations of China (60973048, 61272077, 60974025, 60673101, and 60939003), National 863 Plan Project (2008 AA04Z401, 2009 AA043404), the Natural Science Foundation of Shandong Province (no. Y2007G30), the Scientific and Technological Project of Shandong Province (no. 2007GG3WZ04016), the Science Foundation of Harbin Institute of Technology (Weihai) (HIT (WH) 200807), the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT. NSRIF. 2001120), the China Postdoctoral Science Foundation (2010048 1000), and the Shandong Provincial Key Laboratory of Industrial Control Technique (Qingdao University).